Foundations of Mathematics and Mathematical Practice. The Case of Polish Mathematical School

Authors

DOI:

https://doi.org/10.4467/2543702XSHS.22.007.15973

Keywords:

Polish Mathematical School, logic, logicism, formalism, intuitionism, set theory

Abstract

The foundations of mathematics cover mathematical as well as philosophical problems. At the turn of the 20th century logicism, formalism and intuitionism, main foundational schools were developed. A natural problem arose, namely of how much the foundations of mathematics influence the real practice of mathematicians. Although mathematics was and still is declared to be independent of philosophy, various foundational controversies concerned some mathematical axioms, e.g. the axiom of choice, or methods of proof (particularly, non-constructive ones) and sometimes qualified them as admissible (or not) in mathematical practice, relatively to their philosophical (and foundational) content. Polish Mathematical School was established in the years 1915–1920. Its research program was outlined by Zygmunt Janiszewski (the Janiszewski program) and suggested that Polish mathematicians should concentrate on special branches of studies, including set theory, topology and mathematical logic. In this way, the foundations of mathematics became a legitimate part of mathematics. In particular, the foundational investigations should be conducted independently of philosophical assumptions and apply all mathematically accepted methods, finitary or not, and the same concerns other branches of mathematics. This scientific ideology contributed essentially to the remarkable development of logic, set theory and topology in Poland.

References

Azzouni, Jody 1994: Metaphysical Myths, Mathematical Practice: The Ontology and Epistemology of the Exact Sciences. Cambridge: Cambridge University Press.

Baldwin, John T. 2018: Model Theory and the Philosophy of Mathematical Practice. Formalization without Foundationalism. Cambridge: Cambridge University Press.

Bedürftig, Thomas; Murawski, Roman 2018: Philosophy of Mathematics. Berlin: De Gruyter.

Beth, Evert 1968: The Foundations of Mathematics. A Study in the Philosophy of Science. Amsterdam: North-Holland Publishing Company.

Centrone, Stefania; Kant, Deborah; Sarikaya, Denis 2019 eds.: Reflections on the Foundations of Mathematics. Cham: Springer.

Cevik, Ahmet 2021: Philosophy of Mathematics: Classic and Contemporary Studies. Boca Raton, Fl.: Chapman and Hall/CRC.

Corfield, David 2003: Towards a Philosophy of Real Mathematics. Cambridge: Cambridge University Press.

Dickstein, Samuel 1891: Pojęcia i metody matematyki, Tom pierwszy: Teoria działań (Concepts and Methods of Mathematics. Vol. I: Theory of Operations). Warszawa: Wydawnictwo Redakcyi „Prac matematyczno-fizycznych”.

Duda, Roman 2019: Historia matematyki w Polsce na tle dziejów nauki i kultury (History of Mathematics in Poland against the History of Science and Culture). Warszawa: Instytut Historii Nauki PAN.

Ferreirós, José 2015: Mathematical Knowledge and the Interplay of Practices. Princeton: Princeton University Press.

Frost-Arnold, Greg 2013: Carnap, Tarski and Quine at Harvard. Conversations on Logic, Mathematics and Science. Chicago: Open Court, Chicago.

George, Alexander, Velleman, Daniel 2001: Philosophies of Mathematics. Oxford: Blackwell.

Hamkins, Joel D. 2020: Lectures on the Philosophy of Mathematics. Cambridge, Mass.: Rhe.

Hatcher, William 1982: The Logical Foundations of Mathematics. London: Pergamon Press.

Hilbert, David; Ackermann, Wilhelm 1928: Grundzüge der theoretischen Logik. Berlin: Springer.

Ingarden, Roman 1963: Z badań nad filozofią współczesną (Investigations on Contemporary Philosophy). Warszawa: Państwowe Wydawnictwo Naukowe.

Janiszewski, Zygmunt 1915: Logistyka (Logistics), [in:] Poradnik dla samouków (A Guide for Autodidacts). Warszawa: A. Heflich i S. Michalsk, pp. 449–461.

Janiszewski Zygmunt 1915a: Zagadnienia filozoficzne matematyki” (Philosophical Problems of Mathematics). [In:] Poradnik dla samouków (A Guide for Autodidacts). Warszawa: A. Heflich i S. Michalski, pp. 462–489.

Janiszewski, Zygmunt 1918: O potrzebach matematyki w Polsce” (On Needs of Mathematics in Poland), Nauka Polska, jej potrzeby, organizacja i rozwój I, pp. 11–18.

Irvine, Andrew 2009 ed.: Philosophy of Mathematics. Amsterdam: Elsevier.

Kunen, Kenneth 2011: The Foundations of Mathematics. London: College Publications.

Kuratowski, Kazimierz 1980: A Half Century of Polish Mathematics, Warszawa: Państwowe Wydawnictwo Naukowe.

Kuzawa, Mary G. 1968: Modern Mathematics. The Genesis of a School in Poland. New Haven: College and University Press.

Linnebo, Øystein 2020: Philosophy of Mathematics. Princeton: Princeton University Press.

Lindström, Sten; Palmgren, Erik; Segerberg, Krister; Stoltenberg-HansenViggo 2009 eds.: Logicism, Intuitionism, and Formalism. What Has Become of Them? Berlin: Springer.

Mancosu, Paulo 2008 ed.: The Philosophy of Mathematical Practice, Oxford: Oxford University Press.

McFarland, Andrew; McFarland, Joanna; Smith, James T. 2014: Alfred Tarski. Early Work in Poland – Geometry and Teaching. Berlin: Birkhäuser Verlag.

Mostowski, Andrzej 1955: The Present State of Investigations of the Foundations of Mathematics. Warszawa: Państwowe Wydawnictwo Naukowe.

Mostowski, Andrzej 1964/1979: Thirty Years of Foundational Studies. Lectures on the Development of Mathematical Logic and the Study of the Foundations of Mathematics in 1930–1964. Helsinki Societas Philosophica Fennica; repr. in: A, Mostowski: Foundational Studies: Selected Works, vol. I, Warszawa: Państwowe Wydawnictwo Naukowe 1979, pp. 1–176.

Murawski, Roman 2014: The Philosophy of Mathematics and Logic in the 1920s and 1930s in Poland. Basel: Birkhäuser Verlag.

Murawski, Roman; Woleński, Jan 2008: Andrzej Mostowski on the Foundations and Philosophy of Mathematics. [In:] Andrzej Mostowski and Foundational Studies, ed. by A. Ehrenfeucht; V. W. Marek; M. Srebrny, Amsterdam: IOS Press, pp. 324–337; repr. [in:] R. Murawski: Logos and Máthēma. Studies in the Philosophy and History of Logic. Frankfurt a. M.: Peter Lang, Frankfurt a. M. 2011, pp. 243–257.

Rosello, Joan 2012: From Foundations to Philosophy of Mathematics. An Historical Account of Their Development in XX Century and Beyond. Cambridge: Cambridge Scholars Publishing.

Schirn, Matthias (ed.) 1998: The Philosophy of Mathematics Today. Oxford Clarendon Press.

Shapiro, Stewart 2005: Philosophy of Mathematics: Structure and Ontology. Oxford: Oxford University Press.

Sierpiński, Wacław 1912: Zarys teorii mnogości (Outline of Set Theory). Warszawa: Wydawnictwo Kasy im. Mianowskiego.

Sierpiński, Wacław 1965: Cardinal and Ordinal Numbers. Warszawa: Państwowe Wydawnictwo Naukowe.

Sobociński, Bolesław 1957: In memoriam Jan Łukasiewicz (1878–1956), Philosophical Studies (Maynooth, Ireland) 7, pp. 3–49.

Tarski A. 1930: Fundamentale Begriffe der Methodologie der deduktiven Wissenschaften. Monatshefte dedr Mathematic und Physik 3, pp. 361–404; Eng. tr. in: A. Tarski: Logic, Semantics, Metamathematics, Oxford: Clarendon Press, pp. 60–109.

Tarski, Alfred: 1954: [Remarks of Alfred Tarski], Revue Internationale de Philosophie 27‒28(1‒2), pp. 16‒20; repr. [in:] A. Tarski, Collected Papers, v. III. Basel: Birkhäuser Verlag 1986, pp. 711–714.

Tarski, Alfred 1962: Some Problems and Results Relevant to the Foundations of Set Theory. [In:] Logic, Methodology and Philosophy of Science. Proceedings of the 1960 International Congress, ed. by E. Nagel; P. Suppes; A. Tarski. Stanford: Stanford University Press, pp. 125–135; repr. [in:] A. Tarski, Collected Papers, v. IV. Basel: Birkhäuser Verlag1986, pp. 113–126.

Tarski, Alfred 1992: Drei Briefe an Otto Neurath’ [25. IV. 1930, 10. VI. 1936, 7. IX. 1936]. Grazer Philosophische Studien 43, pp. 1–32.

Taylor, Paul 1999: Practical Foundations of Mathematics. Cambridge: Cambridge University Press.

Woleński, Jan 1995/1999: Mathematical Logic in Poland 1900–1939: People, Circles, Institutions, Ideas. Modern Logic 5, pp. 363–405; repr. in: J. Woleński: Essays in the History of Logic and Logical Philosophy. Kraków: Jagiellonian University Press, Kraków 1999, pp. 59–84. URL: https://projecteuclid.org/journalArticle/Download?urlId=rml%2F1204835573 (accessed on 21 June 2022).

Woleński, Jan 2008: Mathematical Logic in Warsaw: 1918–1939. [In:] Andrzej Mostowski and Foundational Studies, ed. by A. Ehrenfeucht; V. W. Marek; M. Srebrny. Amsterdam: IOS Press, pp. 30–46.

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Published

26-08-2022

How to Cite

Woleński, J. . (2022). Foundations of Mathematics and Mathematical Practice. The Case of Polish Mathematical School. Studia Historiae Scientiarum, 21. https://doi.org/10.4467/2543702XSHS.22.007.15973

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SCIENCE IN POLAND